Theorem efrirr 5611

Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
efrirr	⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		frirr 5607	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴)
2		epelg 5532	. . . 4 ⊢ (𝐴 ∈ 𝐴 → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
3	2	adantl 481	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
4	1, 3	mtbid 324	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴)
5	4	pm2.01da 798	1 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   class class class wbr 5102   E cep 5530   Fr wfr 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-eprel 5531  df-fr 5584

This theorem is referenced by:  tz7.2  5614  ordirr  6338